Question

perform the indicated operations and write the result in standard form.\\(\frac{-12+sqrt{-18}}{60}\\)\\(\frac{-12+sqrt{-18}}{60} = square\\)\\(\text{(type an exact answer, using radicals as needed. type your answer in the form } a + bi.)\\)

Explanation:

Step1: Simplify the imaginary radical

Recall that $\sqrt{-1} = i$. Rewrite $\sqrt{-18}$:
$\sqrt{-18} = \sqrt{18 \times (-1)} = \sqrt{9 \times 2} \times i = 3\sqrt{2}i$

Step2: Substitute back into the expression

Replace $\sqrt{-18}$ with $3\sqrt{2}i$:
$\frac{-12 + 3\sqrt{2}i}{60}$

Step3: Split the fraction

Separate the real and imaginary parts:
$\frac{-12}{60} + \frac{3\sqrt{2}i}{60}$

Step4: Simplify each term

Reduce the fractions to lowest terms:
$\frac{-12}{60} = -\frac{1}{5}$, $\frac{3\sqrt{2}}{60} = \frac{\sqrt{2}}{20}$

Answer:

$-\frac{1}{5} + \frac{\sqrt{2}}{20}i$